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Points and lines in plane

Source: Iran 3rd round 2012-Special Lesson exam-Part1-P1

July 27, 2012
analytic geometrygraphing linesslopecombinatorics proposedcombinatorics

Problem Statement

Prove that the number of incidences of nn distinct points on nn distinct lines in plane is O(n43)\mathcal O (n^{\frac{4}{3}}). Find a configuration for which Ω(n43)\Omega (n^{\frac{4}{3}}) incidences happens.
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